# Tagged Questions

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### Understanding definition of Full Covers

Let $[a, b]$ be a given closed, bounded interval and let $X$ be a subset of $[a,b]$. A collection $\textbf{C}$ of closed subintervals of $[a, b]$ is a full cover of $X$ if to each $x$ in $X$ there ...
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### Ambiguity of Defining Subsequences

Let $(x_n)_{n\in\mathbb{Z}_+}$ be a real sequence such that $x_n=1$ for all $n\in\mathbb{Z}_+$. Consider the sequence $(x_2,x_1,x_3,x_4,x_5,\ldots)$. Argument 1: $(x_2,x_1,x_3,x_4,x_5,\ldots)$ is a ...
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### Definition verification from two different books?

In Kaplansky's Set Theory And Metric Spaces, he mentions a useful example of a neighborhood of $x$ is a closed ball with center $x$. However, one of the theorems in baby Rudin is "Every neighborhood ...
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### Function being continuous at a point

I've been looking at the $\epsilon-\delta$ arguments for determining whether a function is continuous at a point. I'm really stuck on how to choose your $\epsilon$. Specifically lets look at the ...
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### Is it true that $\sum \limits_{i=1}^{\infty} f(i) = \lim_{n \to \infty} \sum \limits_{i=1}^{n} f(i)$?

Is the equality below true? $$\sum \limits_{i=1}^{\infty} f(i) = \lim_{n \to \infty} \sum \limits_{i=1}^{n} f(i)$$
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### How to define $a^x$?

It's so common that we use the function $f(x)=a^x$. But actually how do we define it? In simple language we can say $a^n$ is the number $a$ multiplied with $a$ $n$ times for any $n$ in $\mathbb{N}$ ...
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### Is it true that the definition of an open subset in a metric space is different from the combination of the definitions of subsets and opens sets?

Dear reader of this post, I have a question concerning the equivalence of two definitions of open subsets (in metric spaces). To avoid confusion, I will state the two definitions and then ask my ...
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### Definition of compact set/subset

I found one exercise in my book Let $X$ be a compact subset of a metric space $M$. Prove that $X$ is closed. In the definitions, the book only mentions compact space and never compact set. ...
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### Two questions on $\limsup$: do nested ones commute and sending $n$ to $-\infty$

As for the second question, this is really just me wondering on definitions. For a function $f:\mathbb{R}\to\mathbb{R}$ define $$\limsup_{x\to+\infty}f(x):=\lim_{x\to+\infty}\sup_{z\geq x}f(x).$$ ...
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### For $\lim_{x \rightarrow a}f(x) = L$, if $f(x)$ is not a constant, is $\delta(\epsilon)$ always a monotonically increasing function?

Alternatively, how does the definition of a limit guarantee that if $f(x)$ is not a constant, then a small $\epsilon$ will give me a small $\delta$?
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### Understanding the roots of homomorphism and homeomorphism

I understand the formal (i.e. mathematical) definitions of the terms "homomorphism" and "homeomorphism" as they relate to functions, but I am curious as to the origin of these terms. I don't know ...
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### Example of a (dis)continuous function

The following thought came to my mind: Given we have a function $f$, and for arbitrary $\varepsilon>0$, $f(a+\varepsilon)= 100\,000$ while $f(a) = 1$. Why is or isn't this function continuous? I ...
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### Is uncountably summation defined?

We know that finite and countably summation is defined. But How about uncountably summation, say $$\sum_{i\in \mathbb{R}}0$$ Is it defined?
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### Limit point definition

I have read the definition of a limit point of a set in Real Analysis. The definition goes like: A number $p$ is said to be a limit point of a set of reals, $S$, if every neigbourhood of $p$ has at ...
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### What is the meaning of the expression $\liminf f_n$?

I am a little confused as to what $\liminf f_n$ means for a sequence $f_n$ of functions converging to $f$. I can not locate a definition anywhere.
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### Definition of a metric

I'm having a hard time understanding what the definition of a metric is. From what I think I understand, it's just a method of measurement between $2$ points in $\mathbb R^n$? Is that somewhere along ...
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### Difference between closure and the boundary

I'm having a hard time distinguising the difference between the boundary and the closure of sets. They seem so similar, but that almost sounds too good to be true. So if the boundary is just the ...
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### Improper or Undefined

Let $f(x)=0$ if $x\neq 1$ and $f(1)=\infty$ then the Riemann integral $\int_{0} ^1 f(x)$ $dx$ = $0$ or is it undefined? If we take it as a legitimate function for improper Riemann ...
### How exactly can't $\delta$ depend on $x$ in the definition of uniform continuity?
I'm told that a function defined on an interval $[a,b]$ or $(a,b)$ is uniformly continuous if for each $\epsilon\gt 0$ there exists a $\delta\gt 0$ such that $|x-t|\lt \delta$ implies that ...